C1.5 & C1.6 – Ordering and the Four Operations

Table of Contents

IGCSE Mathematics  |  Core Topic

1. Comparison Symbols (C1.5) #

These symbols are used to compare the size of two numbers or quantities.

Symbol Meaning Example
$=$ is equal to $6 = 6$
$\neq$ is not equal to $3 \neq 7$
$>$ is greater than $8 > 3$
$<$ is less than $2 < 9$
$\geqslant$ is greater than or equal to $x \geqslant 4$   means $x$ can be 4, 5, 6, …
$\leqslant$ is less than or equal to $x \leqslant 10$   means $x$ can be …, 9, 10
Tip: The open (wide) end of $>$ and $<$ always faces the larger number. Think of it like a mouth — it always eats the bigger number:   $8 > 3$.

2. Ordering Numbers by Magnitude (C1.5) #

Magnitude means the size of a number. When ordering numbers, always compare their values carefully — especially with negatives, decimals, and fractions.

Key Rule — Negative Numbers On a number line, values increase from left to right. A number further left is always smaller. $$-5 < -2 < 0 < 3 < 7$$

Ordering Decimals #

Compare digit by digit from left to right. It helps to write all numbers to the same number of decimal places first.

Order from smallest to largest:   $0.35$,   $0.4$,   $0.305$
  1. Write each number to 3 decimal places:   $0.350$,   $0.400$,   $0.305$
  2. Compare:   $305 < 350 < 400$
  3. Answer:   $0.305 < 0.35 < 0.4$

Ordering Fractions #

Convert all fractions to a common denominator before comparing.

Order from smallest to largest:   $\dfrac{2}{3}$,   $\dfrac{3}{4}$,   $\dfrac{5}{6}$
  1. The lowest common denominator of 3, 4 and 6 is 12.
  2. Convert each fraction: $$\frac{2}{3} = \frac{8}{12} \qquad \frac{3}{4} = \frac{9}{12} \qquad \frac{5}{6} = \frac{10}{12}$$
  3. Compare numerators: $8 < 9 < 10$
  4. Answer:   $\dfrac{2}{3} < \dfrac{3}{4} < \dfrac{5}{6}$

3. Negative Numbers — The Four Operations (C1.6) #

Adding and Subtracting #

Rules
  • Adding a negative = subtracting:   $5 + (-3) = 5 – 3 = 2$
  • Subtracting a negative = adding:   $5 – (-3) = 5 + 3 = 8$
Practical Example — Temperature Rise

The temperature at night is $-4\,°\text{C}$. During the day it rises by $11\,°\text{C}$. Find the daytime temperature.

$$-4 + 11 = 7\,°\text{C}$$
Practical Example — Temperature Drop

The temperature is $3\,°\text{C}$ and falls by $8\,°\text{C}$. Find the new temperature.

$$3 – 8 = -5\,°\text{C}$$

Multiplying and Dividing #

Signs Result Example
Positive $\times$ Positive Positive $3 \times 4 = 12$
Negative $\times$ Negative Positive $(-3) \times (-4) = 12$
Positive $\times$ Negative Negative $3 \times (-4) = -12$
Negative $\times$ Positive Negative $(-3) \times 4 = -12$
Remember: The same sign rules apply to division. Same signs → positive.   Different signs → negative.
Calculate $(-6) \div (-2)$

Both signs are negative (same signs) → positive answer.

$$(-6) \div (-2) = 3$$

4. The Four Operations with Fractions (C1.6) #

Important: Always convert mixed numbers to improper fractions before calculating. Convert back at the end if needed.

Adding and Subtracting Fractions #

The denominators must be the same before you can add or subtract.

Calculate $\dfrac{2}{3} + \dfrac{3}{4}$
  1. Find the lowest common denominator (LCD) of 3 and 4:   LCD = 12
  2. Convert both fractions: $$\frac{2}{3} = \frac{8}{12} \qquad \frac{3}{4} = \frac{9}{12}$$
  3. Add the numerators: $$\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$$
  4. Convert to a mixed number:   $\dfrac{17}{12} = 1\dfrac{5}{12}$
Calculate $2\dfrac{1}{2} – 1\dfrac{2}{3}$
  1. Convert to improper fractions:   $2\dfrac{1}{2} = \dfrac{5}{2}$   and   $1\dfrac{2}{3} = \dfrac{5}{3}$
  2. LCD = 6. Convert:   $\dfrac{15}{6} – \dfrac{10}{6}$
  3. Subtract:   $\dfrac{15 – 10}{6} = \dfrac{5}{6}$

Multiplying Fractions #

Multiply the numerators together and the denominators together. Always try to cross-cancel first — this simplifies numbers before you multiply.

Cross-Cancellation Look diagonally across the × sign. If a numerator and the opposite denominator share a common factor, divide both by that factor before multiplying. Write the new (smaller) number above or below the cancelled one.
Calculate $\dfrac{3}{4} \times \dfrac{2}{5}$
  1. Cross-cancel: 4 and 2 share factor 2 → 4 becomes 2, 2 becomes 1
  2. $$\frac{3}{\cancelto{2}{4}} \times \frac{\cancelto{1}{2}}{5} = \frac{3}{2 \times 5} = \frac{3}{10}$$
Calculate $1\dfrac{1}{2} \times 2\dfrac{2}{3}$
  1. Convert to improper fractions:   $\dfrac{3}{2} \times \dfrac{8}{3}$
  2. Cross-cancel: the 3s share factor 3 (both become 1); 2 and 8 share factor 2 (2 becomes 1, 8 becomes 4) $$\frac{\cancelto{1}{3}}{\cancelto{1}{2}} \times \frac{\cancelto{4}{8}}{\cancelto{1}{3}} = \frac{4}{1} = 4$$

Dividing Fractions #

To divide by a fraction, flip the second fraction (find its reciprocal) and then multiply.

Reciprocal The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$.   Example: the reciprocal of $\dfrac{3}{4}$ is $\dfrac{4}{3}$.
Calculate $\dfrac{5}{6} \div \dfrac{2}{3}$
  1. KCF — Keep, Change, Flip:   $\dfrac{5}{6} \div \dfrac{2}{3}$   becomes   $\dfrac{5}{6} \times \dfrac{3}{2}$
  2. Cross-cancel: 6 and 3 share factor 3 → 6 becomes 2, 3 becomes 1 $$\frac{5}{\cancelto{2}{6}} \times \frac{\cancelto{1}{3}}{2} = \frac{5}{4} = 1\frac{1}{4}$$

5. Order of Operations — BIDMAS (C1.6) #

When a calculation has more than one operation, follow this fixed order, remembered as BIDMAS:

Letter Stands For Priority
B Brackets First
I Indices (powers and roots) Second
D / M Division and Multiplication Third — equal priority, left to right
A / S Addition and Subtraction Last — equal priority, left to right
Note: Division and Multiplication have equal priority — work from left to right. The same applies to Addition and Subtraction.
Calculate $3 + 4 \times 2$
  1. Multiplication before addition:   $4 \times 2 = 8$
  2. Then add:   $3 + 8 = 11$

$3 + 4 \times 2 \neq 14$. Do not add first.

Calculate $(3 + 4) \times 2$
  1. Brackets first:   $3 + 4 = 7$
  2. Then multiply:   $7 \times 2 = 14$
Calculate $20 – 3 \times 2^2 + (6 \div 2)$
  1. Brackets:   $6 \div 2 = 3$   →   $20 – 3 \times 2^2 + 3$
  2. Indices:   $2^2 = 4$   →   $20 – 3 \times 4 + 3$
  3. Multiplication:   $3 \times 4 = 12$   →   $20 – 12 + 3$
  4. A/S left to right:   $20 – 12 = 8$,   then $8 + 3 = 11$
Calculate $1.5 + 2.4 \times 3$   (decimals)
  1. Multiplication first:   $2.4 \times 3 = 7.2$
  2. Then add:   $1.5 + 7.2 = 8.7$
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